Abstract

We prove a representation of Aumann-Pettis integrable multivalued martingales by Pettis integrable martingale selectors. Regularity of Aumann-Pettis integrable multivalued martingales and their convergence in Mosco sense, Wijsman topology, and linear topology are established.

Author information

Contact details are reproduced from the original publication and may be historical.

M'hamed El-Louh

Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, Morocco

ellouh.mhamed@gmail.com

Fatima Ezzaki

Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, Morocco

fatimaezzaki@yahoo.fr

Keywords

  • Pettis multivalued martingales
  • regularity
  • representation theorem
  • Mosco convergence
  • linear topology

Mathematics Subject Classification

60G42, 60H05, 58J65

References

  1. F. Akhiat, C. Castaing, F. Ezzaki: Some various convergence results for multivalued martingales, Advances Math. Economics 13 (2010) 1–33.https://doi.org/10.1007/978-4-431-99490-9_1Google Scholar
  2. F. Akhiat, M. El Harami, F. Ezzaki: Pettis conditional expectation of closed convex random sets in a Banach space without RNP, J. Korean Math. Soc. 55(4) (2018) 833–848.Google Scholar
  3. F. Akhiat, F. Ezzaki: Representation theorems for multivalued pramarts, J. Korean Math. Soc. 50(1) (2013) 1–16.https://doi.org/10.4134/JKMS.2013.50.1.001Google Scholar
  4. M. Balcerzak, K. Musial: Vitali type convergence theorems for Banach space valued integrals, Acta Math. Sinica, English Ser. 29(11) (2013) 2027–2036.https://doi.org/10.1007/s10114-013-2578-1Google Scholar
  5. D. Barcenas, W. Urbina: Measurable multifunctions in non separable Banach spaces, SIAM J. Math. Analysis 28(5) (1997) 1212–1226.https://doi.org/10.1137/S0036141095296005Google Scholar
  6. G. Beer: Topologies on Closed and Closed Convex Sets, Kluwed Academic Publishers, Dordrecht (1993).https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  7. B. Cascales, V. Kadets, J. Rodriguez: Measurable selectors and multivalued Pettis integral in non-separable Banach spaces, J. Functional Analysis 256 (2009) 673–699.https://doi.org/10.1016/j.jfa.2008.10.022Google Scholar
  8. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  9. A. Choukairi-Dini: M-Convergence, et régularité des martingales multivoques: Epi-martingales, J. Multivariate Analysis 33 (1990) 49–71.https://doi.org/10.1016/0047-259X(90)90005-3Google Scholar
  10. L. Egghe: Convergence of adapted sequences of Pettis-integrable functions, Pacific J. Math. 114(2) (1984) 345–366.https://doi.org/10.2140/pjm.1984.114.345Google Scholar
  11. K. El Amri, C. Hess: On the Pettis integral of closed valued multifunctions, Set-Valued Analysis 8(4) (2000) 329–360.https://doi.org/10.1023/A:1026547222209Google Scholar
  12. M. El Harami, F. Ezzaki: General Pettis conditional expectation and convergence theorems, Int. J. Math. Statistics 11(1) (2012) 91–111.Google Scholar
  13. C. Hess: On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence, J. Multivariate Analysis 39(1) (1991) 175–201.https://doi.org/10.1016/0047-259X(91)90012-QGoogle Scholar
  14. F. Hiai, H. Umegaki: Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Analysis 7 (1) (1977) 149–182.https://doi.org/10.1016/0047-259X(77)90037-9Google Scholar
  15. D. Q. Luu: Representations and regularity of multivalued martingales, Acta Math. Vietnamica 6(2) (1981) 29–40.Google Scholar
  16. D. Q. Luu: Representation of multivalued regular uniform amarts, Exposé no 9, Séminaire d’Analyse Convexe, Montpellier (1984).Google Scholar
  17. K. Musial: Approximation of Pettis integrable multifunctions with values in arbitrary Banach spaces, J. Convex Analysis 20(3) (2013) 833–870.https://doi.org/10.68381/jca20045Google Scholar
  18. K. Musial: Topics in the theory of Pettis integration, Rendiconti dell’Ist. Mat. dell’Univ. Trieste 23 (1991) 177–262.Google Scholar
  19. J. Neveu: Convergence presque sure des martingales multivoques, Ann. de l’Institut H. Poincaré, Sect. B8 (1972) 1–7.Google Scholar
  20. H. Ziat: Approximation and decomposability in the space of Pettis integrable functions, Int. J. Math. Analysis 6(31) (2012) 1519–1537.Google Scholar
  21. Zhen-Peng Wang, Xing-Hong Xue: On convergence and closedness of multivalued martingales, Trans. Amer. Math. Soc. 341(2) (1994) 807–827.https://doi.org/10.1090/S0002-9947-1994-1154544-XGoogle Scholar

Suggested citation

M. El Allali, M. El-Louh, F. Ezzaki. “On the Convergence and Regularity of Aumann-Pettis Integrable Multivalued Martingales.” Journal of Convex Analysis 27 (2020), No. 4, 1177–1194. https://doi.org/10.68381/jca27062

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2020. Rights now held by Banach Press.